04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 1

EVALUATING LIMITS ANALYTICALLY WITH TRIG

Section 1.3ACalculus AB AP/Dual, Revised ©2013

viet.dang@humble.k12.tx.us

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“0/0” LIMITSAKA: INDETERMINATE FORM

A. Always begin with direct substitutionB. Completely factor the problemC. Simplify and/or Cancel by identifying a function that agrees with for

all x except = . Take the limit of D. Apply algebra rules

1. If necessary, Rationalize the numerator2. Plug in of the function to get the limit

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EXAMPLE 1

Solve

2

4

16lim

4x

x

x

2

4

4 16lim

4 4x

0

0

What form is this?

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EXAMPLE 1

Solve

8

4

4 4lim

4x

x x

x

4

lim 4x

x

4 4

AS X APPROACHES 4, f(x) OR Y APPROACHES 8.

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EXAMPLE 1 (CALCULATOR)

Solve

8

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EXAMPLE 2

Solve

5

4

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YOUR TURN

Solve

1

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WHEN IN ALGEBRA…

You learned to:

NO RADICALS IN THE DENOMINATOR

IN LIMITS, NO RADICALS IN THE NUMERATOR and DENOMINATOR

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EXAMPLE 3

Solve

9

3lim

9x

x

x

9

9 3lim

9 9x

0

0

What form is this?

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EXAMPLE 3

Solve

9

3 3lim

9 3x

x x

x x

9

3 3 9lim

9 3x

x x x

x x

NO NEED TO FOIL THE BOTTOM

9

9lim

9 3x

x

x x

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EXAMPLE 3

Solve

9

9lim

9 3x

x

x x

9

1lim

9 3x

1

3 3

1

6

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EXAMPLE 4

Solve

1

2

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YOUR TURN

Solve Hint: Don’t combine like terms to the denominator, too early

1

4

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EXAMPLE 5

Solve

0

1 15 5lim

x

xx

0

1 15 0 5lim

0x

0

0

What form is this?

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EXAMPLE 5

Solve

0

1 1 lim

5 5 x x

0

1 55lim

5 5 5 5x

x

x x

5

5

5 x 5 x

0

5 5lim

5 5x

x

x

0lim

5 5x

x

x

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EXAMPLE 5

Solve

0

5 5limx

xx

x

0lim

5 5 1x

x x

x

0

1lim

5 5x

x

x x

0

1lim

5 5x x

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EXAMPLE 5

Solve

0

1lim

5 5x x

1

5 5 0

1

25

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EXAMPLE 6

Evaluate

1

4

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YOUR TURN

Solve

1

16

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“SQUEEZE THEOREM”

A. Also known as the “Sandwich theorem,” it is used to evaluate the limit of a function that can't be computed at a given point.

B. For a given interval containing point c, where , , and are three functions that are differentiable and over the interval where is the upper bound and is the lower bound.

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“SQUEEZE THEOREM”

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EXAMPLE 7

Use the Squeeze Theorem to evaluate where c = 1 for

33 2x g x x

3

3

1 1lim3 lim 2x x

x g x x

3

1 1lim3 1 lim 1 2x x

g x

3 3g x

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EXAMPLE 7

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EXAMPLE 8

Use the Squeeze Theorem to evaluate for for which

24 9 4 7x f x x x

7

2

4 4lim 4 9 lim 4 7x x

x f x x x

2

4 4lim 4 4 9 lim 4 4 4 7x x

f x

7 7f x

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YOUR TURN

Use the Squeeze Theorem to evaluate where c = 0 for

9

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SPECIAL TRIGONOMETRIC LIMITS

A. B. C. D. When expressing in radians and not in degrees E. The use help explains the “Squeeze” Theorem

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WHY IS THE LIMIT OF (SIN X)/X, WHEN X APPROACHES 0 EQUAL TO 1?

0

sinlim 1x

x

x

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WHY IS THE LIMIT OF (1 – COS X)/X, WHEN X APPROACHES 0 EQUAL TO 0?

0

1 coslim 0x

x

x

MEMORIZE IT!

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EXAMPLE 9

Solve

0

tanlimx

x

x

0

sinlim

cosx

x

x x

Is there another way of rewriting tan (x)?

Split the fraction up so we can isolate and utilize a trigonometric limit

0

sin 1lim

cosx

x

x x

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EXAMPLE 9

0

sin 1lim

cosx

x

x x

0 0

sin 1lim lim

cosx x

x

x x

Utilize the Product Property of Limits

1

1cos 0

1

Solve

1

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EXAMPLE 10

Solve

0

sin 4limx

x

x

0

sin 4 4lim

4x

x

x

Try to convert it to one of its trig limits.

0

sin 44 lim

4x

x

x

4 4 1

Try to get it where the sine trig function to cancel. Whatever is applied to the bottom, must be applied to the top.

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EXAMPLE 10

Solve

4

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EXAMPLE 11

Solve

2

3

1 sin 2

3

x

x

0

1 sin 2 2lim

3 2x

x

x

0

2 sin 2lim

3 2x

x

x

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YOUR TURN

Solve

5

3

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PATTERN?

Solve = 4Solve = Solve = Solve = Solve =

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EXAMPLE 12

Solve

2

0

1 coslimx

x

x

0

1 cos 1 coslim

1x

x x

x

Split the fraction up so we can isolate and utilize a trigonometric limit

0 0

1 coslim lim 1 cosx x

xx

x

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EXAMPLE 13

Solve

0 0

1 coslim lim 1 cosx x

xx

x

0 1 cos 0

0 2

cos(0) = 1

0

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EXAMPLE 14

Solve

20

sin sin coslimx

x x x

x

0

sin 1 coslimx

x x

x x

1 0

0

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YOUR TURN

Solve

0

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AP MULTIPLE CHOICE PRACTICE QUESTION (NON-CALCULATOR)

Solve

(A) π(B) 1(C) 0(D) –1

D

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AP MULTIPLE CHOICE PRACTICE QUESTION (NON-CALCULATOR)

Solve

(A) 0(B) –π/2(C) (2√2)/π(D) 2/π

D

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AP FREE RESPONSE PRACTICE QUESTION (NON-CALCULATOR)

If a ≠ 0, then determine . If the limit does not exist, explain why.

2

1

2a

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ASSIGNMENT

Page 6727-36 all, 63-71 all, 73, 89